Representation of the slowness model

As with the ray tracing method, I assume that the slowness model is smoothly varying everywhere except along the specified interfaces where slowness discontinuities appear. To trace a ray within a grid cell in the smooth slowness region, we need to specify the slowness values in the cell on the basis of the given slowness samples at four corners of the cell. The constant slowness cell is the simplest one, but it causes problems in the amplitude calculation because of the slowness discontinuities formed around the boundaries of the cell. The constant gradient cell has the same potential problem as the constant slowness cell if the gradient of the slowness model is large. A simple continuous slowness model can be constructed by using the bilinear interpolation. The bilinear interpolation within a cell is defined as follows:

$$m(\alpha,\beta)=(1-\alpha)(1-\beta)m_{ij}+\alpha(1-\beta)m_{i+1j}+ (1-\alpha)\beta m_{ij+1}+\alpha \beta m_{i+1j+1},$$ (8)

where $0 \le \alpha \le 1$ and $0 \le \beta \le 1$.The components of the slowness gradient within the cell are

$$\left\{ \begin{array} {lll} \displaystyle{\partial m \over \... ...laystyle{m_{i+1j+1}-m_{i+1j} \over \Delta z},\end{array}\right.$$ (9)

which are not constant, and thus may complicate local ray tracing algorithm. Further numerical experiments are required to determine which type of cells should be used.

If a slowness cell contains a portion of an interface, we need to specify the dip of the interface as well as the slownesses at two sides of the interface.